Actions of compact groups on compact \((4, m)\)-quadrangles (Q1591228)

In the present work (which is an expanded version of a part of his PhD thesis), Biller characterizes certain compact generalized quadrangles by the size of their automorphism groups. Both the results and the methods introduced by Biller constitute in the reviewer's opinion a major milestone in topological geometry. Generalized quadrangles are buildings of rank 2 and type \(C_2\). A fundamental theorem by \textit{J. Tits} [Buildings of spherical type and finite BN-pairs, Lect. Notes Math. 386 (1974; Zbl 0295.20047)] says that all irreducible spherical buildings of rank at least 3 satisfy the Moufang condition and are in some sense `known'. This is not true for buildings of rank 2, and a general classification is impossible. One of the aims of incidence geometry is therefore to characterize these rank 2 buildings either by geometric properties (configurations) or by properties of their automorphism group. Here, the most important classes of geometries are the finite ones (related to finite group theory or combinatorics) and compact geometries (related for example to Lie groups, boundaries of Hadamard spaces, or valuations), which are the subject of the present article. The `simplest' class of rank two buildings are the projective planes, and compact projective planes have been studied with great success by Salzmann and his school over the last forty years [see \textit{H. Salzmann} et al., `Compact projective planes'. Berlin: de Gruyter (1996; Zbl 0851.51003)]. A result of \textit{N. Knarr} [Forum Math. 2, No. 6, 603-612 (1990; Zbl 0711.51002)] and the reviewer [Diss. Tübingen (1994; Zbl 0844.51006)] says that such a compact (connected and finite dimensional) building is of type \(A_2\) (a projective plane), \(C_2\) (a generalized quadrangle) or \(G_2\) (a generalized hexagon). Among these, the generalized quadrangles are the most difficult class, since there are infinitely many examples of arbitrarily high dimension. Due to the fact that there is no bound on the dimension (as for projective planes), new ideas are needed. Biller's main tool is a careful application of results about the cohomology of abelian transformation groups, which are combined with results on full subquadrangles by \textit{H. Van Maldeghem} and the reviewer [Geom. Dedicata 78, No. 3, 279-300 (1999; Zbl 0942.51007)]. In this way, Biller obtains information about the maximum rank of a compact automorphism group, and this leads to an upper bound for the maximum dimension of a compact automorphism group. The resulting bounds are particularly good for \((4,m)\)-quadrangles (the numbers refer to the dimensions of the panels) and lead to a complete characterization of the classical \((4,m)\)-quadrangles in terms of the dimensions of compact groups of automorphisms. This paper complements in a nice way the results by \textit{T. Grundhöfer}, \textit{N. Knarr}, and the reviewer [Geom. Dedicata 55, No. 1, 95-114 (1995; Zbl 0840.51009) and Geom. Dedicata 83, No. 1-3, 1-29 (2000; Zbl 0974.51014)], where the classical compact polygons are characterized by the transitivity of their automorphism group. Biller's work will certainly be important for any further research in this area.